A073336 - OEIS

%I #22 Aug 18 2019 07:30:53

%S 0,1,2,4,8,13,21,33,51,76,111,159,226,315,435,593,805,1077,1435,1893,

%T 2486,3237,4198,5405,6935,8843,11235,14201,17893,22437,28052,34929,

%U 43371,53653,66201,81410,99876,122155,149063,181399,220280,266811,322524,388960

%N Total number of square parts in all partitions of n.

%H Vaclav Kotesovec, <a href="/A073336/b073336.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from Alois P. Heinz)

%F a(n) = Sum_{k=1..n} A046951(k)*A000041(n-k).

%F G.f.: Sum_{i>=1} x^(i^2)/(1 - x^(i^2)) / Product_{j>=1} (1 - x^j). - _Ilya Gutkovskiy_, Jan 24 2017

%p b:= proc(n, i) option remember; `if`(n=0, [1, 0],

%p       `if`(i<1, [0, 0], add((l->l+[0, `if`(j>0 and issqr(i),

%p        l[1]*j, 0)])(b(n-i*j, i-1)), j=0..iquo(n, i))))

%p     end:

%p a:= n-> b(n, n)[2] :

%p seq(a(n), n=0..60);  # _Alois P. Heinz_, Feb 19 2013

%t b[n_, i_] :=  b[n, i] = If[n == 0, {1, 0}, If[i<1, {0, 0}, Sum[Function[{l}, l+{0, If[j>0 && IntegerQ[Sqrt[i]], l[[1]]*j, 0]}][b[n-i*j, i-1]], {j, 0, Quotient[n, i]}]]]; a[n_] := b[n, n][[2]]; Table[a[n], {n, 0, 60}] (* _Jean-François Alcover_, May 13 2015, after _Alois P. Heinz_ *)

%Y Cf. A000041, A046951, A309535.

%K easy,nonn

%O 0,3

%A _Vladeta Jovovic_, Aug 22 2002

%E More terms from _Emeric Deutsch_, Nov 18 2004

%E a(0) inserted by _Alois P. Heinz_, Feb 19 2013